The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping

The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.


Introduction
The study of dynamical system is closely related to some important problems in natural science (such as turbulence in fluid mechanics, three-body problem in celestial mechanics, etc.), which attract a large number of natural scientists to study for a long time. However, the content of general research is limited to the case of finite dimension. With the development of science and technology, especially the rapid development of computer technology, it is already possible to learn more about the evolution and final state of infinite dimensional dynamical systems through computers. Since the 1980s, the infinite dimensional dynamical system has been studied in detail, such as the Russian mathematician O. A. La-dyzhenskoya ( [1] [2] [3]), French mathematician R. Temam ([4] [5]), American mathematician G. Sell [6] and Guo Boling [7] who is an academician of the Chinese. They have made a deep research on a kind of infinite dimensional dynamical system generated by a class of nonlinear development equations with dissipative effects. Under certain conditions, it is proved that all these systems have a global attractor. Furthermore, the upper and lower bounds of the Hausdorff dimension and Fractal dimension of the global attractor are estimated. Many monographs have been published in this field, see ([8] [9] [10] [11]). Igor et al. [8] considered the long-time behavior of solutions to a damped wave equation with a critical source term and investigated the existence and various properties of global attractors. In [9], Yang and Wang considered the longtime behavior of solution for the following Kirchhoff type equation with a strong dissipation: They proved that the related continuous semigroup ( ) S t possesses in the phase space with low regularity a global attractor that is connected. Kirchhoff type differential equations are a kind of classical problems in partial differential equations. In 1883, German physicist G. Kirchhoff [12] established the equation when he studied the vibration of strings.
is the lateral displacement under space coordinate x and time coordinate t, E the Young modulus, ρ the mass density, h the cross-sectional area, L the length, p 0 the initial axial tension, f the external force. It corrects the classic D'alembert wave equation. Thus, the process of string vibration is described more precisely. This model has been widely used in non-newtonian fluid mechanics, astrophysics, image processing, plasma problems and elastic theory. Early research on the Kirchhoff type equations could be found in the literature ( [13]- [20]).
Wu and Tsai [21] studied the initial boundary value problem of the following Kirchhoff-type beam equation They prove that the existence and uniqueness of the global solution and the decay estimation.
In the process of vibration and deformation of the vibration system, the characteristic that the amplitude of the vibration gradually decreases due to the inherent reasons of the system or the interaction with the outside world is called damping, and mathematically called dissipation.
Igor Chueshov [22] studied long-time dynamics of a class of quasilinear wave equations with a strong damping term They proved the existence and uniqueness of the weak solutions and studied their properties for a wide class of nonlinearities which covers the case of possible degeneration (or even negativity) of the stiffness coefficient and the case of a supercritical source term. They also established the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals.
Recently, Guoguang Lin, Zhuoxi Li [23] studied the initial boundary value problem for a class of high order Kirchhoff type equations with nonlinear non-local source term and strongly damped term a nonlinear non-local source term.
They proved that the existence of the family of global attractors and estimated their Hausdorff dimensions and Fractal dimensions.
In the present paper, we deal with the following the higher-order nonlinear Kirchhoff type problem involving a strong damping term where 1 m > is a positive integer, Ω is a bounded domain with smooth homogeneous Dirichlet boundary ∂Ω in ( ) 1 n R n ≥ , ν represents the unit normal vector directed towards the exterior of Ω . D represents gradient operator, which means is a strong damping term, here β is a positive constant. M is a non-negative function that satisfies some conditions. When 2 p = , is a bounded absorbtion set. In order to obtain our results, we consider system (1)-(3) under some assumptions on ( ) M s , ε and p. Preclsely, we state the general assumptions: The remainder of this article is organized as follows: In Sect. 2, we prove the existness and uniqueness of the family of global attractors and in Sect. 3, the estimate of the upper bound of Hausdorff dimension and Fractal dimension for the family of global attractors have been obtained.   We have

The Existence and Uniqueness of the Family of Global Attractors
By using Holder's inequality, Young's inequality and Poincare's inequality to process the items in Equation (4) one by one, we get ( ) where 1 λ is the first eigenvalue of −∆ in Ω with homogeneous Dirichlet boundary condition, and all of the following are this definition.
Dealing with the second term in Equation (4), we get There are two cases to estimate the above equation Intergrating the above inequality to get Dealing with the third term in Equation (4), we get ( ) By using Young's inequality and Poincare's inequality to deal with the strong damping term, we have

Modern Nonlinear Theory and Application
By using Young's inequality to deal with external force term, we get Combining with (4)-(9), we have There exists a smooth and global solu- Proof. Taking the scalar product in H of Equation (1) with By using Holder's inequality, Young's inequality and Poincare's inequality to process the items in (10) one by one, we get From (H 1 ), a proof method that similar to Lemma 2.1 can be obtained, Dealing with the third term in Equation (10), we get By using Young's inequality and Poincare's inequality to deal with the strong damping term, we have By using Young's inequality to deal with the external force term, we get Substituting (11)-(15) into (10), we receive The proof is complete. The general conclusions about the system of nonlinear ordinary differential equations ensure that the solution of problems It can be seen that the priori estimate of Lemma 2.1 and Lemma 2.2 by Equation (17) and Equation (18) . According to the Rellich -Kohdrachov compact embedding theorem, we arrive at In Equation (16), we make l µ = and take limit. For fixed j and j µ ≥ , we  We obtain By multiplying (19) by where ζ is between (2) There exists a bounded absorbing set 0 B is the bounded absorbing set in k E and satisfies where ( ) S t is the solution semigroup that generated by problem (1) (2) Furthermore, for any ( ) B is the bounded absorbing set of ( )


The proof is complete.

The Estimate of the Upper Bound of Hausdorff Dimension and Fractal Dimension for the family of Global Attractors
First, the Equation (1)   Proof. Assume ( ) where ς is between in m D u and m D u .

Conclusion
In this paper, a class of high order Kirchhoff type equation has been investigated. In recent years, much work concerning the low order Kirchhoff type equation has been published. However, to the best of my knowledge, there were few long-time behaviors for the high order Kirchhoff type equation with strong damping. We have proved the existence and uniqueness of the global solution of the problem by using prior estimates and the Galerkin's method under proper assumptions for the rigid term. Furthermore, we have been obtained the Hausdorff dimension and Fractal dimension of the family of global attractors.